An explicit formula of the limit of the heat kernel measures on the spheres embedded in $\mathbb {R}^\infty$

TL;DR

This paper derives an explicit formula for the limit of heat kernel measures on high-dimensional spheres, showing their convergence to a Gaussian measure in infinite-dimensional space.

Contribution

It provides a precise explicit formula for the limiting Gaussian measure of heat kernel measures on spheres as dimension tends to infinity.

Findings

Heat kernel measures on spheres converge to Gaussian measures in infinite dimensions.

An explicit formula for the limiting Gaussian measure is derived.

The convergence occurs under proper scaling of the sphere radius.

Abstract

We show that the heat kernel measures based at the north pole of the spheres $S^{N-1}(\sqrt N)$, with properly scaled radius $\sqrt N$ and adjusted center, converge to a Gaussian measure in $\mathbb R^\infty$, and find an explicit formula for this measure.